aggregating risk capital, with an application to

Introduction Duality Applications Conclusions and Extensions

Aggregating Risk Capital, with an Application toOperational Risk

2007 version

Paul Embrechts1 Giovanni Puccetti2

1Department of Mathematics ETH Zurich, CH-8092 Zurich, Switzerland

2Department of Mathematics for Decisions, University of Firenze, 50134 Firenze, Italy

available atwww.dmd.unifi.it/puccetti

P. Embrechts and G. Puccetti ETHZ Zurich, DMD Firenze

Bounding Risk Measures


The present talk is mainly based on the following papers:

JMVA Embrechts, P. and G. Puccetti (2006). Bounds for functions ofmultivariate risks.J. Mult. Analysis, 97(2), 526–547.

F&S Embrechts, P. and G. Puccetti (2006b). Bounds for functionsofdependent risks.Finance Stoch. 10(3), 341–352.

GRIR Embrechts, P. and G. Puccetti (2006c). Aggregating risk capital,with an application to operational risk.Geneva Risk. Insur.Rev., 31(2), 71–90.




The problem at hand

The problem at hand

On some probability space (Ω,A, P), consider a random vector

X := (X1, . . . ,Xn)

of n one-period financial losses or insurance claims,

and fix its marginal dfsF1, . . . ,Fn.

Thejoint df of the random vectorXis not completely determined by theFi’s.




The problem at hand

There are infinitely many distributions for the vectorX which areconsistent with the initial choice of the marginals.

Figure:Two bivariate dfs havingN(0, 1)-marginals and the same correlation

Given an aggregating functionψ : Rn → R, which is the df giving theworst-possible Value-at-Risk (VaR) for ψ(X) ?




Value-at-Risk

The Value-at-Risk at probability levelα for ψ(X) is the maximumaggregate loss which can occur with probabilityα, α ∈ [0, 1].

If G (the df ofψ(X)) is strictly increasing,VaRα(ψ(X)) is the unique thresholdt at whichF(t) = α, i.e. F−1(α).




Value-at-Risk

Searching for the worst-possible VaR means looking for

mψ(s) := inf P[ψ(X) < s] : Xi v Fi, i = 1, . . . , n, s ∈ R.

Indeed, according to the definition of VaR, we have

VaRα(ψ(X)) ≤ m−1ψ (α), α ∈ [0, 1].




Duality theorem

mψ(s) is a linear problem over a convex feasible space of measures.Therefore, it admits adual representation.

Main Duality Theorem (Ruschendorf (1982))

mψ(s) = inf P[ψ(X) < s] : Xi v Fi, i = 1, . . . , n

= 1− inf

n∑

i=1

∫

fidFi : fi ∈ L1(Fi), i ∈ N s.t.

n∑

i=1

fi(xi) ≥ 1[s,+∞)(ψ(x)) for all x ∈ Rn

.




Duality theorem

Some remarks on the dual problem

mψ(s), as well as its dual counterpart, is very difficult to solve.Solutions are known only in few cases.

• Whenn = 2; see Ruschendorf (1982).

• Whenn > 2, the only explicit solution we know is givenin Ruschendorf (1982) for the sum of risks uniformly distributedon the unit interval

Even if we do not solve the dual problem,dual admissible functions provide bounds on the solutions which

are conservative from a risk management viewpoint




Standard/Dual bounds

We callstandard bounds those bounds obtained by choosingpiecewise-constant dual choices.

• Standard bounds are those typically obtained from elementaryprobability; see: Denuit, Genest, and Marceau (1999). Theyaresharp whenn = 2.

We calldual bounds those bounds obtained by choosingpiecewise-linear dual choices.

• Dual bounds arebetter than standard bounds whenn > 2 but areavailable only forψ = +.

• Extensions are stated for portfolios of vectors; see [JMVA]




Application 1

Bounds on Value-at-Risk

VaRα(∑3

i=1 Xi), exact VaRα(∑3

i=1 Xi), upper bound

α independence comonoton. dual standard

0.90 7.54 8.85 14.44 15.380.95 9.71 12.73 19.50 20.630.99 16.06 25.16 35.31 37.030.999 29.78 53.99 69.98 73.81

Table:Range for VaR for a Log-Normal(-0.2,1)-portfolio.




Application 1

Bounds on Value-at-Risk (large portfolios!)

VaRα(∑10

i=1 Xi) VaRα(∑100

i=1 Xi) VaRα(∑1000

i=1 Xi)

α dual standard dual standard dual standard

0.90 0.669 1.485 11.039 149.850 150.162 14998.5000.95 1.353 2.985 22.227 229.850 301.823 29998.5000.99 2.985 14.985 111.731 1499.850 1515.111 149998.5000.999 68.382 149.985 1118.652 14999.850 15164.604 1499998.500

Table:Upper bounds for VaRα(∑n

i=1 Xi) of three Pareto portfolios of differentdimensions. Data in thousands.




Application 2

Application 2: Aggregation of OpRisk losses

Under the New Basel Capital Accord (Basel II) banks are required toset aside capital for the specific purpose of offsetting OpRisk.

OpRisk The risk of losses resulting from inadequate or failedinternal processes, people andsystems, or external events.

Included islegal risk, excluded are strategic/business and reputationalrisk.




Application 2

OpRisk pillar 1 issue: Loss Distribution Approach (LDA)

• Operational lossesLi,j are separately modeled ineight businesslines (rows) and byseven risk types (columns) in the 56-cellBasel matrix

• Marginal risks havenon-homogeneous distributions

• Capital requirement to be calculated as the sum of VaR1 year0.999

across the cells of the matrix:∑8

i=1∑7

j=1 VaR0.999(Li,j)




Application 2

In standard practice (see Moscadelli (2004)), the OR capital chargecan be calculated aggregating risks BL-wise yielding capital estimates

VaR1., . . . ,VaR8.

RT1 . . . RTj . . . RT7

BL1 → VaR1....

......

BLi Li,j → VaRi....

......

BL8 → VaR8.

Finally calculate (as indicated in Basel II) thecomonotonic value:VaRR =

∑8i=1 VaRi.

How reliable is this procedure?




Application 2

α Basel II value dual bound standard bound

0.99 2.8924× 104 1.4778× 105 2.6950× 105

0.995 6.7034× 104 3.3922× 105 6.1114× 105

0.999 4.8347× 105 2.3807× 106 4.1685× 106

0.9999 8.7476× 106 4.0740× 107 6.7936× 107

Table:Range for VaRα(

∑8i=1 BLi

)

for the OR portfolio given in Moscadelli(2004).

• The computation of standard bounds is non-trivial in thenon-homogeneous case; see [GRIR]

• The dual bound is prudential, more realistic and economicallyadvantageous with respect to the standard one.

• There is no mathematical reason to drop the worst-case bounds if nodependence assumptions on the portfolio are explicitly made.




Application 2

Recall thatVaRR =∑8

i=1 VaRi.

The OR capital charge could be calculated aggregating risksalsoRT-wise, yielding the capital estimateVaRC =

∑7j=1 VaR.j

RT1 . . . RTj . . . RT7

BL1 → VaR1....

......

BLi Li,j → VaRi....

......

BL8 → VaR8.

↓ . . . ↓ . . . ↓

VaR.1 . . . VaR.j . . . VaR.7

• VaRC, VaRR?

• Which factors does∆ :=VaRC- VaRR depend upon?




Application 2

A toy model approach

• Simplified 2× 3 Basel matrix

• Li,j ∼ Pareto(4) for alli = 1, 2, j = 1, 2, 3

• Dependence among theLi,j is modeled by a six-dimensionalGumbel copula with parameterθ

• the Gumbel copula is symmetric and has Gumbel projections,losses are identically distributed

↓

The only asymmetry in OR aggregation is caused by the fact thatthe Basel matrix is not square




Application 2

Conclusions from the toy model approach

θ VaRC VaRR ∆

1.00 independence 18.31 14.46 3.851.10 dependence 21.32 19.38 1.941.25 higher dependence 23.71 22.62 1.09+∞ comonotonicity 27.74 27.74 0

• Large differences between the two VaR-aggregation methods.

• Starting from a maximum in the independence set-up,∆

becomes smaller as the strength of dependence increases.

• Under the comonotonic assumption,∆ = 0 due to VaR additivity.

• Coming soon: a more sophisticatedsoft model




Conclusions

Conclusions

• The worst-possible VaR for a non-decreasing function ofdependent risks can be calculated when the portfolio istwo-dimensional.

• When dealing with more than two risks, the problem gets muchmore complicated and we provide adual bound which we proveto be better than the standard one generally used in the literature.

• OpRisk VaR-aggregation leads to problems anddiversification effects have to be handled with care.




Final Conclusions and possible extensions

Extensions (more research is needed!)

• Exact VaR bounds whenn > 2, calculation of bounds for otherportfolio functionsψ

• Basel II has some issues to solve (2008+)

• Problems of scaling when fixing marginal dfs(Market+ Credit+ Op Risk).

• For a textbook treatment, see

Visit the book zone:www.ma.hw.ac.uk/˜mcneil/book/index.html




Final Conclusions and possible extensions

Acknowledgements

The second author would like to thank

RiskLab, ETH Zurich

for financial support.



References

For Further Reading I

Denuit, M., C. Genest, andE. Marceau (1999). Stochastic bounds on sums of dependent risks.Insurance Math. Econom. 25(1), 85–104.

Dutta, K. and J. Perry (2006). A tale of tails: an empirical analysis of loss distribution modelsfor estimating operational risk capitalWorking Papers, Federal Reserve Bank of Boston

Embrechts, P., A. Hoing, and A. Juri (2003). Using copulae to bound the Value-at-Risk forfunctions of dependent risks.Finance Stoch. 7(2), 145–167.

Embrechts, P., A. Hoing, and G. Puccetti (2005). Worst VaR scenariosInsurance Math.Econom., 37(1), 115–134.

Moscadelli, M. (2004). The modelling of operational risk: experience with the analysis of thedata collected by the Basel Committee. Preprint, Banca d’Italia.

Ruschendorf, L. (1982). Random variables with maximum sums. Adv. in Appl. Probab. 14(3),623–632.

Williamson, R. C. and T. Downs (1990). Probabilistic arithmetic. I. Numerical methods for

calculating convolutions and dependency bounds.Internat. J. Approx. Reason. 4(2), 89–158.



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